Publications

Published

Quotients of d-frames (with Achim Jung, Aleš Pultr), Applied Categorical Structures, 2019.

BibTeX

@article{jakljungpultr2019,
  author="Jakl, Tom{\'a}{\v{s}} and Jung, Achim and Pultr, Ale{\v{s}}",
  title="Quotients of d-Frames",
  journal="Applied Categorical Structures",
  year="2019",
  month="Jun",
  day="01",
  volume="27",
  number="3",
  pages="261--275",
  issn="1572-9095",
  doi="10.1007/s10485-018-09553-7"
}

Abstract

It is shown that every d-frame admits a complete lattice of quotients. Quotienting may be triggered by a binary relation on one of the two constituent frames, or by changes to the consistency or totality structure, but as these are linked by the reasonableness conditions of d-frames, the result in general will be that both frames are factored and both consistency and totality are increased.

d-Frames as algebraic duals of bitopological spaces, Charles University and University of Birmingham (Ph.D. thesis), 2018.

Free constructions and coproducts of d-frames (with Achim Jung), Proceedings of CALCO 2017.

BibTeX

@InProceedings{jakljung2017,
  author =    {Tom{\'a}{\v{s}} Jakl and Achim Jung},
  title =     {{Free Constructions and Coproducts of d-Frames}},
  booktitle = {7th Conference on Algebra and Coalgebra in Computer Science (CALCO 2017)},
  pages =     {14:1--14:15},
  series =    {Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =      {978-3-95977-033-0},
  ISSN =      {1868-8969},
  year =      {2017},
  volume =    {72},
  editor =    {Filippo Bonchi and Barbara K{\"o}nig},
  publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =   {Dagstuhl, Germany},
  doi =       {10.4230/LIPIcs.CALCO.2017.14},
  annote =    {Keywords: Free construction, d-frame, coproduct, C-ideals}
}

Abstract

A general theory of presentations for d-frames does not yet exist. We review the difficulties and give sufficient conditions for when they can be overcome. As an application we prove that the category of d-frames is closed under coproducts.

Bitopology and four-valued logic (with Achim Jung, Aleš Pultr), Proceedings of the 32nd Annual Conference on Mathematical Foundations of Programming Semantics (MFPS XXXII), Electronic Notes in Theoretical Computer Science, 2016.

BibTeX

@article{jakljungpultr2016,
  title = "Bitopology and Four-valued Logic",
  author = "Tom{\'a}{\v{s}} Jakl and Achim Jung and Ale{\v{s}} Pultr",
  journal = "Electronic Notes in Theoretical Computer Science",
  volume = "325",
  pages = "201 - 219",
  year = "2016",
  note = "The Thirty-second Conference on the Mathematical Foundations of Programming Semantics (MFPS XXXII)",
  issn = "1571-0661",
  doi = "https://doi.org/10.1016/j.entcs.2016.09.039",
  keywords = "Bilattices, d-frames, nd-frames, bitopological spaces, four-valued logic"
}

Abstract

Bilattices and d-frames are two different kinds of structures with a four-valued interpretation. Whereas d-frames were introduced with their topological semantics in mind, the theory of bilattices has a closer connection with logic. We consider a common generalisation of both structures and show that this not only still has a clear bitopological semantics, but that it also preserves most of the original bilattice logic. Moreover, we also obtain a new bitopological interpretation for the connectives of four-valued logic.

Tightness relative to some (co)reflections in topology (with Richard N. Ball, Bernhard Banaschewski, Aleš Pultr, Joanne Walters-Waylande), Quaestiones Mathematicae, 2016.

BibTeX

@article{bbjpw2016,
  author =  {Richard N. Ball and Bernhard Banaschewski and
             Tom{\'a}{\v{s}} Jakl and Ale{\v{s}} Pultr and
             Joanne Walters-Wayland},
  title =   {Tightness relative to some (co)reflections in topology},
  journal = {Quaestiones Mathematicae},
  volume =  {39},
  number =  {3},
  pages =   {421-436},
  year  =   {2016},
  publisher = {Taylor & Francis},
  doi =       {10.2989/16073606.2015.1073191}
}

Abstract

We address what might be termed the reverse reflection problem: given a monoreflection from a category A onto a subcategory B, when is a given object b ∈ B the reflection of a proper subobject? We start with a well known specific instance of this problem, namely the fact that a compact metric space is never the Čech-Stone compactification of a proper subspace. We show that this holds also in the pointfree setting, i.e., that a compact metrizable locale is never the Čech-Stone compactification of a proper sublocale. This is a stronger result than the classical one, but not because of an increase in scope; after all, assuming weak choice principles, every compact regular locale is the topology of a compact Hausdorff space. The increased strength derives from the conclusion, for in general a space has many more sublocales than subspaces. We then extend the analysis from metric locales to the broader class of perfectly normal locales, i.e., those whose frame of open sets consists entirely of cozero elements. We include a second proof of these results which is purely algebraic in character.

At the opposite extreme from these results, we show that an extremally disconnected locale is a compactification of each of its dense sublocales. Finally, we analyze the same phenomena, also in the pointfree setting, for the 0-dimensional compact reflection and for the Lindelöf reflection.

In preparation

  1. Tomáš Jakl: Canonical extensions of locally compact frames, submitted.
  2. Mai Gehrke, Tomáš Jakl, Luca Reggio: On duality and limits of finite structures.
  3. Tomáš Jakl, Achim Jung: Free constructions of d-frames.

Other

Also see my Google Scholar, Github and Genealogy profiles.

My ORCID is 0000-0003-1930-4904.

Web Analytics